Simple Harmonic Motion

Page 1

Figure 1

When a force acting on a body is directly proportional to the displacement of that body from its resting (equilibrium) position and when this force is in the opposite direction of the displacement from the resting position, then the body will move with what it called simple harmonic motion. Simple harmonic motion is a type of periodic motion. A spring with a mass attached can be used to observe simple harmonic motion. When the spring and the mass are held vertically so that gravity pulls the mass toward the ground, the end of the spring is displaced toward the ground. A restoring force will act to pull the spring and the mass attached back toward the resting position according to Hooke’s Law. The relation of the force F to the displacement x from the equilibrium position is: đ??š = −đ?‘˜đ?‘Ľ

(1)


Where: k = The spring constant The spring with the mass will move up and down periodically. The motion is described by the following equation: (2) đ?‘Ľ = đ??´ cos 2đ?œ‹đ?‘“đ?‘Ą Where: A = The amplitude of the motion f = The frequency of the motion The period of the motion is the amount of time it takes to repeat the period of motion once. It is related to the spring constant and the size of the mass (m, measured in kg): đ?‘š (3) đ?‘‡ = 2đ?œ‹âˆš đ?‘˜ and can also be expressed as the inverse of the frequency of the motion: đ?‘‡=

1 đ?‘“

(4)

In this experiment we examine the motion of a mass attached to a spring and oscillating vertically. The force acting on the spring and the position of the mass are measured simultaneously.

einstein™Tablet+ with MiLAB or Android/iOS Tablet with MiLAB and einstein™LabMate 

Force sensor



Distance sensor



A mass attached to a spring (the frequency of the oscillation should be 0.5-2 Hz and the amplitude should be 5-20 cm)



1 kg mass (2)



Stand with a clamp to hold the force sensor and the spring



Stand with a clamp to hold the distance sensor

C-clamp to clamp the stand to the counter Meter stick 1 large 5cm x 8cm index card Balance to measure the mass

1.

Launch MiLAB (

).

2. 3. 4.

Connect the Force sensor to one of the ports on the einstein™Tablet+ or einstein™LabMate. Connect the Distance sensor to one of the ports on the einstein™Tablet+ or einstein™LabMate. Assemble the equipment as shown in Figure 1: a. Hang the spring from the Force sensor. b. Carefully attach the 1 kg mass to the spring.


c.

5.

Place the Distance sensor directly beneath the mass. When the spring is fully extended, the mass and the Distance sensor must be at least 40 cm apart. Make sure that only the Force and Distance sensors are selected.

Program the sensor to log data according to the following setup: Force Sensor

Force, Pull - positive ( 50 N) (N)

Set As Zero

ON

Distance Sensor

Distance (m)

Rate:

25/sec

Duration:

40 Sec

1.

Using a meter stick and two known masses, measure the spring constant k: a. Place a 1 kg mass of the end of the spring. Gently let it come to rest. (Do not allow the spring to oscillate). b. When the spring comes to its resting (equilibrium) position, use the meter stick to measure the distance between the floor and the bottom of the 1 kg mass. Record this distance. c. Put an additional 1 kg mass on the spring and let it come gently to rest. (Do this carefully!) When the spring comes to its equilibrium position, use the meter stick to measure the distance between the floor and the bottom of the first 1 kg mass used. Find the difference in the spring extensions measured.

Note: It is necessary to measure to the bottom of the first 1 kg mass in order to determine the distance the spring stretched. Measurements should always be taken relative to the same point on the first mass.

2.

Calculate and record the change in Force, ΔF, using Newton’s second law of motion F = ma. In the case of using two 1 kg masses, this is given by 1 kg x 9.8 m.s2. e. Calculate the spring constant k using Hooke’s Law F = kx. Divide the change in force ΔF by the change in displacement, Δx, of the first 1 kg mass after the second 1 kg mass was added. Using a 1 kg mass, measure the period of oscillation (T) of the mass on the end of the spring: a. Place the 1 kg mass on the end of the spring, suspended above the Distance sensor. b. Attach the 5 cm x 8 cm index card to the bottom of the mass so that the broad side of the card is facing the Distance sensor. Gently pull the 1 kg mass down and let go. It will oscillate up and down.

3.

Tap Run (

4.

After about 15 oscillations, tap Stop (

5.

Save your data by tapping Save (

d.

) to begin recording data. ).

) on the upper toolbar.

For more information on working with graphs see: Working with Graphs in MiLAB.


2.

Working with either the Force curve or the Distance curve, measure the period of oscillation by using two cursors. Put the first cursor on the first peak and the second cursor on the 11th peak. Record the value of Δt, which will appear in the text box on the graph as dx. The period, T, will be Δt/10. Record this value of T. Calculate the spring constant k using the period of oscillation and the following rearrangement of Equation (3): 4đ?œ‹ 2 đ?‘š đ?‘˜= đ?‘‡2

3. 4.

Compare the values of k which you measured using two different methods. Export the data ( ) as a .csv file.

5. 6. 7.

Create a plot of force vs distance. Fit a straight line to the resulting graph. From Equation (1), the slope of this graph is the spring constant k. Compare this value of k with those calculated previously.

Force (N)

1.

Figure 2

1. 2.

Measure the period of oscillation with a 2 kg mass and calculate the spring constant k. Explore the relationship of the velocity to the distance: a. Use the distance sensor to measure the velocity of the oscillating spring. b. Calculate the amplitude, A for the maximum velocity: 2đ?œ‹ đ?‘‰đ?‘šđ?‘Žđ?‘Ľ = đ??´ đ?‘‡


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